The Wave Equation with Acoustic Boundary Conditions on Non-Locally Reacting Surfaces

The aim of the paper is to study the problem { u t t − c 2 Δ u = 0 a m p ; in R × Ω , μ v t t − d i v Γ ( σ ∇ Γ v ) + δ v t + κ v + ρ u t = 0 a m p ; on R × Γ 1 , v t = ∂ ν u a m p ; on R × Γ 1 , ∂ ν u = 0 a m p ; on R × Γ 0 , u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) a m p ; in Ω , v ( 0 , x ) = v 0 ( x ) , v t ( 0 , x ) = v 1 ( x ) a m p ; on Γ 1 , {equation*} {cases} uₜₜ-c^2Δ u=0 &in R× Ω,\\ μ vₜₜ- div_Γ(σ ∇ _Γ v)+δ v_t+κ v+ρ u_t =0 &on R× Γ ₁,\\ v_t =∂ _ν u &on R× Γ ₁,\\ ∂ _ν u=0 &on R× Γ ₀,\\ u(0,x)=u_0(x), u_t(0,x)=u_1(x) & in Ω,\\ v(0,x)=v_0(x), v_t(0,x)=v_1(x) & on Γ ₁, {cases} {equation*} where Ω Ω is a open domain of R N R^N with uniformly C r C^r boundary ( N ≥ 2 N≥ 2 , r ≥ 1 r≥ 1 ), Γ = ∂ Ω Γ =∂ Ω , ( Γ 0 , Γ 1 ) (Γ _0,Γ _1) is a relatively open partition of Γ Γ with Γ 0 Γ _0 (but not Γ 1 Γ _1 ) possibly empty. Here d i v Γ div_Γ and ∇ Γ ∇ _Γ denote the Riemannian divergence and gradient operators on Γ Γ , ν ν is the outward normal to Ω Ω , the coefficients μ , σ , δ , κ , ρ μ ,σ ,δ , κ , ρ are suitably regular functions on Γ 1 Γ _1 with ρ , σ ρ ,σ and μ μ uniformly positive while c c is a positive constant. This problem have been proposed long time ago by Beale and Rosencrans, when N = 3 N=3 , σ = 0 σ =0 , r = ∞ r=∞ , ρ ρ is constant, κ , δ ≥ 0 κ ,δ ≥ 0 , to model acoustic wave propagation with locally reacting boundary. In this paper we first study well-posedness in the natural energy space and give regularity results. Hence we give precise qualitative results for solutions when Ω Ω is bounded and r = 2 r=2 , ρ ρ is constant, κ , δ ≥ 0 κ ,δ ≥ 0 . These results motivate a detailed discussion of the derivation of the problem in Theoretical Acoustics and the consequent proposal of adding to the model the integral condition ∫ Ω u t = c 2 ∫ Γ 1 v ∫ _Ω u_t=c^2∫ Γ _1v .